Editing Central limit theorem (section)

==Beyond the classical framework==
Asymptotic normality, that is, [[Convergence in distribution|convergence]] to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.

===Convex body===
{{math theorem | math_statement = There exists a sequence {{math|''ε<sub>n</sub>'' ↓ 0}} for which the following holds. Let {{math|''n'' ≥ 1}}, and let random variables {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} have a [[Logarithmically concave function|log-concave]] [[Joint density function|joint density]] {{mvar|f}} such that {{math|1=''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') = ''f''({{abs|''x''<sub>1</sub>}}, ..., {{abs|''x<sub>n</sub>''}})}} for all {{math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}},  and {{math|1=E(''X''{{su|b=''k''|p=2}}) = 1}} for all {{math|1=''k'' = 1, ..., ''n''}}. Then the distribution of

<math display="block"> \frac{X_1+\cdots+X_n}{\sqrt n} </math>

is {{mvar|ε<sub>n</sub>}}-close to <math display="inline"> \mathcal{N}(0, 1)</math> in the [[Total variation distance of probability measures|total variation distance]].{{sfnp|Klartag|2007|loc=Theorem 1.2}}}}

These two {{mvar|ε<sub>n</sub>}}-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

Another example: {{math|1=''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') = const · exp(−({{abs|''x''<sub>1</sub>}}<sup>''α''</sup> + ⋯ + {{abs|''x<sub>n</sub>''}}<sup>''α''</sup>)<sup>''β''</sup>)}} where {{math|''α'' > 1}} and {{math|''αβ'' > 1}}. If {{math|1=''β'' = 1}} then {{math|''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}} factorizes into {{math|const · exp (−{{abs|''x''<sub>1</sub>}}<sup>''α''</sup>) … exp(−{{abs|''x<sub>n</sub>''}}<sup>''α''</sup>), }} which means {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} are independent. In general, however, they are dependent.

The condition {{math|1=''f''(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') = ''f''({{abs|''x''<sub>1</sub>}}, ..., {{abs|''x<sub>n</sub>''}})}} ensures that {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} are of zero mean and [[uncorrelated]];{{Citation needed|date=June 2012}} still, they need not be independent, nor even [[Pairwise independence|pairwise independent]].{{Citation needed|date=June 2012}} By the way, pairwise independence cannot replace independence in the classical central limit theorem.{{sfnp|Durrett|2004|loc=Section 2.4, Example 4.5}}

Here is a [[Berry–Esseen theorem|Berry–Esseen]] type result.

{{math theorem | math_statement = Let {{math|''X''<sub>1</sub>, ..., ''X<sub>n</sub>''}} satisfy the assumptions of the previous theorem, then{{Sfnp|Klartag|2008|loc=Theorem 1}}

<math display="block"> \left| \mathbb{P} \left( a \le \frac{ X_1+\cdots+X_n }{ \sqrt n } \le b \right) - \frac1{\sqrt{2\pi}} \int_a^b e^{-\frac{1}{2} t^2} \, dt \right| \le \frac{C}{n} </math>

for all {{math|''a'' < ''b''}}; here {{mvar|C}} is a [[mathematical constant|universal (absolute) constant]]. Moreover, for every {{math|''c''<sub>1</sub>, ..., ''c<sub>n</sub>'' ∈ '''R'''}} such that {{math|1=''c''{{su|b=1|p=2}} + ⋯ + ''c''{{su|b=''n''|p=2}} = 1}},

<math display="block"> \left| \mathbb{P} \left( a \le c_1 X_1+\cdots+c_n X_n \le b \right) - \frac{1}{\sqrt{2\pi}} \int_a^b e^{-\frac{1}{2} t^2} \, dt \right| \le C \left( c_1^4+\dots+c_n^4 \right). </math>}}

The distribution of {{math|{{sfrac|''X''<sub>1</sub> + ⋯ + ''X<sub>n</sub>''|{{sqrt|''n''}}}}}} need not be approximately normal (in fact, it can be uniform).{{sfnp|Klartag|2007|loc=Theorem 1.1}} However, the distribution of {{math|''c''<sub>1</sub>''X''<sub>1</sub> + ⋯ + ''c<sub>n</sub>X<sub>n</sub>''}} is close to <math display="inline"> \mathcal{N}(0, 1)</math> (in the total variation distance) for most vectors {{math|(''c''<sub>1</sub>, ..., ''c<sub>n</sub>'')}} according to the uniform distribution on the sphere {{math|1=''c''{{su|b=1|p=2}} + ⋯ + ''c''{{su|b=''n''|p=2}} = 1}}.

===Lacunary trigonometric series===
{{math theorem | name = Theorem ([[Raphaël Salem|Salem]]–[[Antoni Zygmund|Zygmund]]) | math_statement =
Let {{mvar|U}} be a random variable distributed uniformly on {{math|(0,2π)}}, and {{math|1=''X<sub>k</sub>'' = ''r<sub>k</sub>'' cos(''n<sub>k</sub>U'' + ''a<sub>k</sub>'')}}, where
* {{mvar|n<sub>k</sub>}} satisfy the lacunarity condition: there exists {{math|''q'' > 1}} such that {{math|''n''<sub>''k'' + 1</sub> ≥ ''qn''<sub>''k''</sub>}} for all {{mvar|k}},
* {{mvar|r<sub>k</sub>}} are such that<br /><math display="block"> r_1^2 + r_2^2 + \cdots = \infty \quad\text{ and }\quad \frac{ r_k^2 }{ r_1^2+\cdots+r_k^2 } \to 0, </math>
* {{math|0 ≤ ''a''<sub>''k''</sub> < 2π}}.
Then<ref name=Zygmund/>{{sfnp|Gaposhkin|1966|loc=Theorem 2.1.13}}

<math display="block"> \frac{ X_1+\cdots+X_k }{ \sqrt{r_1^2+\cdots+r_k^2} } </math>

converges in distribution to <math display="inline"> \mathcal{N}\big(0, \frac{1}{2}\big)</math>.}}

===Gaussian polytopes===
{{math theorem | math_statement =
Let {{math|''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub>}} be independent random points on the plane {{math|'''R'''<sup>2</sup>}} each having the two-dimensional standard normal distribution. Let {{mvar|K<sub>n</sub>}} be the [[convex hull]] of these points, and {{mvar|X<sub>n</sub>}} the area of {{mvar|K<sub>n</sub>}} Then{{sfnp|Bárány|Vu|2007|loc=Theorem 1.1}}

<math display="block"> \frac{ X_n - \operatorname E (X_n) }{ \sqrt{\operatorname{Var} (X_n)} } </math>
converges in distribution to <math display="inline"> \mathcal{N}(0, 1)</math> as {{mvar|n}} tends to infinity.}}

The same also holds in all dimensions greater than 2.

The [[convex polytope|polytope]] {{mvar|K<sub>n</sub>}} is called a Gaussian random polytope.

A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.{{sfnp|Bárány|Vu|2007|loc=Theorem 1.2}}

===Linear functions of orthogonal matrices===
A linear function of a matrix {{math|'''M'''}} is a linear combination of its elements (with given coefficients), {{math|'''M''' ↦ tr('''AM''')}} where {{math|'''A'''}} is the matrix of the coefficients; see [[Trace (linear algebra)#Inner product]].

A random [[orthogonal matrix]] is said to be distributed uniformly, if its distribution is the normalized [[Haar measure]] on the [[orthogonal group]] {{math|O(''n'','''R''')}}; see [[Rotation matrix#Uniform random rotation matrices]].

{{math theorem | math_statement = Let {{math|'''M'''}} be a random orthogonal {{math|''n'' × ''n''}} matrix distributed uniformly, and {{math|'''A'''}} a fixed {{math|''n'' × ''n''}} matrix such that {{math|1=tr('''AA'''*) = ''n''}}, and let {{math|1=''X'' = tr('''AM''')}}. Then<ref name=Meckes/> the distribution of {{mvar|X}} is close to <math display="inline"> \mathcal{N}(0, 1)</math> in the total variation metric up to{{clarify|reason=what does up to mean here|date=June 2012}} {{math|{{sfrac|2{{sqrt|3}}|''n'' − 1}}}}.}}

===Subsequences===
{{math theorem | math_statement = Let random variables {{math|''X''<sub>1</sub>, ''X''<sub>2</sub>, ... ∈ ''L''<sub>2</sub>(Ω)}} be such that {{math|''X<sub>n</sub>'' → 0}} [[Weak convergence (Hilbert space)|weakly]] in {{math|''L''<sub>2</sub>(Ω)}} and {{math|''X''{{su|b=''n''|2}} → 1}} weakly in {{math|''L''<sub>1</sub>(Ω)}}. Then there exist integers {{math|''n''<sub>1</sub> < ''n''<sub>2</sub> < ⋯}} such that

<math display="block"> \frac{ X_{n_1}+\cdots+X_{n_k} }{ \sqrt k }</math>

converges in distribution to <math display="inline"> \mathcal{N}(0, 1)</math> as {{mvar|k}} tends to infinity.{{sfnp|Gaposhkin|1966|loc=Sect. 1.5}}}}

===Random walk on a crystal lattice===

The central limit theorem may be established for the simple [[random walk]] on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.<ref>{{cite book |last1=Kotani |first1=M. |last2=Sunada |first2=Toshikazu |author-link2=Toshikazu Sunada |date=2003 |title=Spectral geometry of crystal lattices |publisher=Contemporary Math |volume=338 |pages=271–305 |isbn=978-0-8218-4269-0}}</ref><ref>{{cite book |author-link=Toshikazu Sunada |last=Sunada |first=Toshikazu |date=2012 |title=Topological Crystallography – With a View Towards Discrete Geometric Analysis|series=Surveys and Tutorials in the Applied Mathematical Sciences |volume=6 |publisher=Springer |isbn=978-4-431-54177-6}}</ref>